By Stavros N. Busenberg

Differential Equations and functions in Ecology, Epidemics, and inhabitants Problems.

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**Example text**

We now consider the stability properties of this equilibrium. N(Xjt) If -Τί(χ). 3), we define u(x,t) N(x,t) A straight-forward computation reveals that (ignoring higher order terms) ^~(x3t)=K 9t = 2 ύ 9 J u(x3t) - ^(xit)+vlLl-Ti(x)']u(x, 9 a? 1) David Green, Jr. and Harlan W. 2) > 0) (xzl, 5 s < 0). 4) J _f V(0) = ψΓπ; = 0 has a nontrivial solution. 5) have no eigenvalue λ ε 0 with real part >0. The sufficiency of this condition follows from the results in [ll] for partial functional differential equations in the C = {<$>:[-T30~\ ■> L (03i\)\§ phase space is continuous}.

A o Where 0 < Θ < 2π, a,b,o,d,k,1 are positive constants 2 and e(Q) is a C -function on the circle. Solving for a steady-state solution yields H (Q) o V ° = ΤΤω ; " e(Q) I ( c ~ el J + c dQ2 ( Thus the effect of diffusion near a maximum of reduce p n(®) H Q n( ) ' i-s to below the value for the undiffused system and clearly if H (Q) Pn(Q) J e(Q) has a sharp enough maximum the value of becomes negative so that too great a safety for the prey leads to extinction of predators near such points in this case.

Denotes situations where (due to numerical inaccuracy) Diffusion and Hereditary Effects in a Class of Population Models 27 it was not possible to determine whether solutions were slowly decaying to tion close to N N. 210. N*(x>t) for K = 2, r = 10, See [4] for a more detailed discussion of the simu- lation. V. SUMMARY For a population model incorporating the effects of delay- ed self-regulation and diffusion it is shown that when the diffusion coefficient, rate of growth, v, K, is larger than the "intrinsic" all solutions (regardless of the specific form of the delay) must approach zero.